3.218 \(\int \frac {\sin ^8(c+d x)}{(a-b \sin ^4(c+d x))^2} \, dx\)

Optimal. Leaf size=320 \[ \frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 b^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 b^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\tan ^5(c+d x)}{4 b d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\tan (c+d x)}{4 b d (a-b)}+\frac {x}{b^2} \]

[Out]

x/b^2+1/8*a^(1/4)*arctan((a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b^(3/2)/d/(a^(1/2)-b^(1/2))^(3/2)-1/8*a^(
1/4)*arctan((a^(1/2)+b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b^(3/2)/d/(a^(1/2)+b^(1/2))^(3/2)-1/2*a^(1/4)*arctan((
a^(1/2)-b^(1/2))^(1/2)*tan(d*x+c)/a^(1/4))/b^2/d/(a^(1/2)-b^(1/2))^(1/2)-1/2*a^(1/4)*arctan((a^(1/2)+b^(1/2))^
(1/2)*tan(d*x+c)/a^(1/4))/b^2/d/(a^(1/2)+b^(1/2))^(1/2)-1/4*tan(d*x+c)/(a-b)/b/d+1/4*tan(d*x+c)^5/b/d/(a+2*a*t
an(d*x+c)^2+(a-b)*tan(d*x+c)^4)

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Rubi [A]  time = 0.45, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {3217, 1313, 1275, 12, 1122, 1166, 205, 1287, 203} \[ \frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 b^{3/2} d \left (\sqrt {a}-\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}-\sqrt {b}}}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 b^{3/2} d \left (\sqrt {a}+\sqrt {b}\right )^{3/2}}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 b^2 d \sqrt {\sqrt {a}+\sqrt {b}}}+\frac {\tan ^5(c+d x)}{4 b d \left ((a-b) \tan ^4(c+d x)+2 a \tan ^2(c+d x)+a\right )}-\frac {\tan (c+d x)}{4 b d (a-b)}+\frac {x}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^2,x]

[Out]

x/b^2 - (a^(1/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] - Sqrt[b]]*b^2*d) + (
a^(1/4)*ArcTan[(Sqrt[Sqrt[a] - Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*(Sqrt[a] - Sqrt[b])^(3/2)*b^(3/2)*d) - (a^(
1/4)*ArcTan[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(2*Sqrt[Sqrt[a] + Sqrt[b]]*b^2*d) - (a^(1/4)*ArcT
an[(Sqrt[Sqrt[a] + Sqrt[b]]*Tan[c + d*x])/a^(1/4)])/(8*(Sqrt[a] + Sqrt[b])^(3/2)*b^(3/2)*d) - Tan[c + d*x]/(4*
(a - b)*b*d) + Tan[c + d*x]^5/(4*b*d*(a + 2*a*Tan[c + d*x]^2 + (a - b)*Tan[c + d*x]^4))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 1122

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[(d^3*(d*x)^(m - 3)*(a + b*
x^2 + c*x^4)^(p + 1))/(c*(m + 4*p + 1)), x] - Dist[d^4/(c*(m + 4*p + 1)), Int[(d*x)^(m - 4)*Simp[a*(m - 3) + b
*(m + 2*p - 1)*x^2, x]*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b^2 - 4*a*c, 0] && Gt
Q[m, 3] && NeQ[m + 4*p + 1, 0] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1275

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Simp[(f*
(f*x)^(m - 1)*(a + b*x^2 + c*x^4)^(p + 1)*(b*d - 2*a*e - (b*e - 2*c*d)*x^2))/(2*(p + 1)*(b^2 - 4*a*c)), x] - D
ist[f^2/(2*(p + 1)*(b^2 - 4*a*c)), Int[(f*x)^(m - 2)*(a + b*x^2 + c*x^4)^(p + 1)*Simp[(m - 1)*(b*d - 2*a*e) -
(4*p + 4 + m + 1)*(b*e - 2*c*d)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[
p, -1] && GtQ[m, 1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

Rule 1287

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rule 1313

Int[(((f_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_))/((d_.) + (e_.)*(x_)^2), x_Symbol] :> -Dist
[f^4/(c*d^2 - b*d*e + a*e^2), Int[(f*x)^(m - 4)*(a*d + (b*d - a*e)*x^2)*(a + b*x^2 + c*x^4)^p, x], x] + Dist[(
d^2*f^4)/(c*d^2 - b*d*e + a*e^2), Int[((f*x)^(m - 4)*(a + b*x^2 + c*x^4)^(p + 1))/(d + e*x^2), x], x] /; FreeQ
[{a, b, c, d, e, f}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1] && GtQ[m, 2]

Rule 3217

Int[sin[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]^4)^(p_.), x_Symbol] :> With[{ff = FreeF
actors[Tan[e + f*x], x]}, Dist[ff^(m + 1)/f, Subst[Int[(x^m*(a + 2*a*ff^2*x^2 + (a + b)*ff^4*x^4)^p)/(1 + ff^2
*x^2)^(m/2 + 2*p + 1), x], x, Tan[e + f*x]/ff], x]] /; FreeQ[{a, b, e, f}, x] && IntegerQ[m/2] && IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\sin ^8(c+d x)}{\left (a-b \sin ^4(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^8}{\left (1+x^2\right ) \left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^4 \left (a+a x^2\right )}{\left (a+2 a x^2+(a-b) x^4\right )^2} \, dx,x,\tan (c+d x)\right )}{b d}-\frac {\operatorname {Subst}\left (\int \frac {x^4}{\left (1+x^2\right ) \left (a+2 a x^2+(a-b) x^4\right )} \, dx,x,\tan (c+d x)\right )}{b d}\\ &=\frac {\tan ^5(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int -\frac {2 a b x^4}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{8 a b^2 d}-\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{b \left (1+x^2\right )}+\frac {a \left (1+x^2\right )}{b \left (a+2 a x^2+(a-b) x^4\right )}\right ) \, dx,x,\tan (c+d x)\right )}{b d}\\ &=\frac {\tan ^5(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{b^2 d}-\frac {a \operatorname {Subst}\left (\int \frac {1+x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{b^2 d}-\frac {\operatorname {Subst}\left (\int \frac {x^4}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{4 b d}\\ &=\frac {x}{b^2}-\frac {\tan (c+d x)}{4 (a-b) b d}+\frac {\tan ^5(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}-\frac {\left (a \left (1-\frac {\sqrt {b}}{\sqrt {a}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 b^2 d}-\frac {\left (a \left (1+\frac {\sqrt {b}}{\sqrt {a}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{2 b^2 d}+\frac {\operatorname {Subst}\left (\int \frac {a+2 a x^2}{a+2 a x^2+(a-b) x^4} \, dx,x,\tan (c+d x)\right )}{4 (a-b) b d}\\ &=\frac {x}{b^2}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^2 d}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^2 d}-\frac {\tan (c+d x)}{4 (a-b) b d}+\frac {\tan ^5(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}+\frac {\left (\sqrt {a} \left (\sqrt {a}+\sqrt {b}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{8 \left (\sqrt {a}-\sqrt {b}\right ) b^{3/2} d}+\frac {\left (\sqrt {a} \left (2 \sqrt {a}-\frac {a+b}{\sqrt {b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a-\sqrt {a} \sqrt {b}+(a-b) x^2} \, dx,x,\tan (c+d x)\right )}{8 (a-b) b d}\\ &=\frac {x}{b^2}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}-\sqrt {b}} b^2 d}+\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}-\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \left (\sqrt {a}-\sqrt {b}\right )^{3/2} b^{3/2} d}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {\sqrt {a}+\sqrt {b}} b^2 d}-\frac {\sqrt [4]{a} \tan ^{-1}\left (\frac {\sqrt {\sqrt {a}+\sqrt {b}} \tan (c+d x)}{\sqrt [4]{a}}\right )}{8 \left (\sqrt {a}+\sqrt {b}\right )^{3/2} b^{3/2} d}-\frac {\tan (c+d x)}{4 (a-b) b d}+\frac {\tan ^5(c+d x)}{4 b d \left (a+2 a \tan ^2(c+d x)+(a-b) \tan ^4(c+d x)\right )}\\ \end {align*}

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Mathematica [A]  time = 5.02, size = 262, normalized size = 0.82 \[ \frac {-\frac {\sqrt {a} \left (4 \sqrt {a}+5 \sqrt {b}\right ) \tan ^{-1}\left (\frac {\left (\sqrt {a}+\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}+a}}\right )}{\left (\sqrt {a}+\sqrt {b}\right ) \sqrt {\sqrt {a} \sqrt {b}+a}}+\frac {2 a b (\sin (4 (c+d x))-6 \sin (2 (c+d x)))}{(a-b) (8 a+4 b \cos (2 (c+d x))-b \cos (4 (c+d x))-3 b)}+\frac {\sqrt {a} \left (4 \sqrt {a}-5 \sqrt {b}\right ) \tanh ^{-1}\left (\frac {\left (\sqrt {a}-\sqrt {b}\right ) \tan (c+d x)}{\sqrt {\sqrt {a} \sqrt {b}-a}}\right )}{\left (\sqrt {a}-\sqrt {b}\right ) \sqrt {\sqrt {a} \sqrt {b}-a}}+8 (c+d x)}{8 b^2 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Sin[c + d*x]^8/(a - b*Sin[c + d*x]^4)^2,x]

[Out]

(8*(c + d*x) - (Sqrt[a]*(4*Sqrt[a] + 5*Sqrt[b])*ArcTan[((Sqrt[a] + Sqrt[b])*Tan[c + d*x])/Sqrt[a + Sqrt[a]*Sqr
t[b]]])/((Sqrt[a] + Sqrt[b])*Sqrt[a + Sqrt[a]*Sqrt[b]]) + (Sqrt[a]*(4*Sqrt[a] - 5*Sqrt[b])*ArcTanh[((Sqrt[a] -
Sqrt[b])*Tan[c + d*x])/Sqrt[-a + Sqrt[a]*Sqrt[b]]])/((Sqrt[a] - Sqrt[b])*Sqrt[-a + Sqrt[a]*Sqrt[b]]) + (2*a*b
*(-6*Sin[2*(c + d*x)] + Sin[4*(c + d*x)]))/((a - b)*(8*a - 3*b + 4*b*Cos[2*(c + d*x)] - b*Cos[4*(c + d*x)])))/
(8*b^2*d)

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fricas [B]  time = 1.33, size = 3544, normalized size = 11.08 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^2,x, algorithm="fricas")

[Out]

1/32*(32*(a*b - b^2)*d*x*cos(d*x + c)^4 - 64*(a*b - b^2)*d*x*cos(d*x + c)^2 - 32*(a^2 - 2*a*b + b^2)*d*x + ((a
*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(-((a^3*b^4
- 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^
7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2
)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(32*a^3 - 166*a^2*b + 1125/4*a*b^2 - 625/4*b^3 - 1/4*(128*a^
3 - 664*a^2*b + 1125*a*b^2 - 625*b^3)*cos(d*x + c)^2 + 1/2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 +
3*b^9)*d^3*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4
*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (24*a^3*b^2 - 127*a^2*b^
3 + 220*a*b^4 - 125*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((6
4*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10
+ 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*
d^2)) + 1/4*(2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*cos(d*x + c)^2 - (16*a^4*b^3 -
73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2)*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 62
5*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))) - ((a*b^3 -
b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(-((a^3*b^4 - 3*a^
2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a
^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3
*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(32*a^3 - 166*a^2*b + 1125/4*a*b^2 - 625/4*b^3 - 1/4*(128*a^3 - 664
*a^2*b + 1125*a*b^2 - 625*b^3)*cos(d*x + c)^2 - 1/2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 + 3*b^9)
*d^3*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 -
20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))*cos(d*x + c)*sin(d*x + c) - (24*a^3*b^2 - 127*a^2*b^3 + 220
*a*b^4 - 125*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(-((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 -
464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^
2*b^11 - 6*a*b^12 + b^13)*d^4)) + 16*a^3 - 47*a^2*b + 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2)) +
1/4*(2*(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*cos(d*x + c)^2 - (16*a^4*b^3 - 73*a^3*
b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2)*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4
)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))) + ((a*b^3 - b^4)*d*
cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(((a^3*b^4 - 3*a^2*b^5 +
3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 +
15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 16*a^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3
*a^2*b^5 + 3*a*b^6 - b^7)*d^2))*log(-32*a^3 + 166*a^2*b - 1125/4*a*b^2 + 625/4*b^3 + 1/4*(128*a^3 - 664*a^2*b
+ 1125*a*b^2 - 625*b^3)*cos(d*x + c)^2 + 1/2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 + 3*b^9)*d^3*sq
rt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*
b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))*cos(d*x + c)*sin(d*x + c) + (24*a^3*b^2 - 127*a^2*b^3 + 220*a*b^4
- 125*b^5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4
*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 -
6*a*b^12 + b^13)*d^4)) - 16*a^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2)) + 1/4*(2*
(16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*cos(d*x + c)^2 - (16*a^4*b^3 - 73*a^3*b^4 + 12
3*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2)*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*
b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))) - ((a*b^3 - b^4)*d*cos(d*x
+ c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b^3 + b^4)*d)*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6
- b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*
b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4)) - 16*a^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5
+ 3*a*b^6 - b^7)*d^2))*log(-32*a^3 + 166*a^2*b - 1125/4*a*b^2 + 625/4*b^3 + 1/4*(128*a^3 - 664*a^2*b + 1125*a
*b^2 - 625*b^3)*cos(d*x + c)^2 - 1/2*(2*(2*a^4*b^5 - 9*a^3*b^6 + 15*a^2*b^7 - 11*a*b^8 + 3*b^9)*d^3*sqrt((64*a
^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 1
5*a^2*b^11 - 6*a*b^12 + b^13)*d^4))*cos(d*x + c)*sin(d*x + c) + (24*a^3*b^2 - 127*a^2*b^3 + 220*a*b^4 - 125*b^
5)*d*cos(d*x + c)*sin(d*x + c))*sqrt(((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2*sqrt((64*a^5 - 464*a^4*b + 124
1*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^1
2 + b^13)*d^4)) - 16*a^3 + 47*a^2*b - 35*a*b^2)/((a^3*b^4 - 3*a^2*b^5 + 3*a*b^6 - b^7)*d^2)) + 1/4*(2*(16*a^4*
b^3 - 73*a^3*b^4 + 123*a^2*b^5 - 91*a*b^6 + 25*b^7)*d^2*cos(d*x + c)^2 - (16*a^4*b^3 - 73*a^3*b^4 + 123*a^2*b^
5 - 91*a*b^6 + 25*b^7)*d^2)*sqrt((64*a^5 - 464*a^4*b + 1241*a^3*b^2 - 1450*a^2*b^3 + 625*a*b^4)/((a^6*b^7 - 6*
a^5*b^8 + 15*a^4*b^9 - 20*a^3*b^10 + 15*a^2*b^11 - 6*a*b^12 + b^13)*d^4))) - 8*(a*b*cos(d*x + c)^3 - 2*a*b*cos
(d*x + c))*sin(d*x + c))/((a*b^3 - b^4)*d*cos(d*x + c)^4 - 2*(a*b^3 - b^4)*d*cos(d*x + c)^2 - (a^2*b^2 - 2*a*b
^3 + b^4)*d)

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giac [B]  time = 1.17, size = 1563, normalized size = 4.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^2,x, algorithm="giac")

[Out]

1/8*((2*(6*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 21*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b
)*a^2*b + 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 3*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(
a*b)*b^3)*(a*b^2 - b^3)^2*abs(-a + b) - (12*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^5*b^2 - 63*sqrt(a^2 - a*b -
sqrt(a*b)*(a - b))*a^4*b^3 + 116*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a^3*b^4 - 86*sqrt(a^2 - a*b - sqrt(a*b)*(
a - b))*a^2*b^5 + 16*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*a*b^6 + 5*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*b^7)*ab
s(-a*b^2 + b^3)*abs(-a + b) - (9*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^4 - 51*sqrt(a^2 - a*b - s
qrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^5 + 102*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*b^6 - 82*sqrt(a^2
- a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^7 + 17*sqrt(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^8 + 5*sqrt
(a^2 - a*b - sqrt(a*b)*(a - b))*sqrt(a*b)*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) + arctan(tan(d*x + c
)/sqrt((a^2*b^2 - a*b^3 + sqrt((a^2*b^2 - a*b^3)^2 - (a^2*b^2 - a*b^3)*(a^2*b^2 - 2*a*b^3 + b^4)))/(a^2*b^2 -
2*a*b^3 + b^4))))/((3*a^7*b^4 - 21*a^6*b^5 + 59*a^5*b^6 - 85*a^4*b^7 + 65*a^3*b^8 - 23*a^2*b^9 + a*b^10 + b^11
)*abs(-a*b^2 + b^3)) - (2*(6*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3 - 21*sqrt(a^2 - a*b + sqrt(a*b)
*(a - b))*sqrt(a*b)*a^2*b + 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a*b^2 + 3*sqrt(a^2 - a*b + sqrt(a
*b)*(a - b))*sqrt(a*b)*b^3)*(a*b^2 - b^3)^2*abs(-a + b) + (12*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^5*b^2 - 63
*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^4*b^3 + 116*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a^3*b^4 - 86*sqrt(a^2 -
a*b + sqrt(a*b)*(a - b))*a^2*b^5 + 16*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*a*b^6 + 5*sqrt(a^2 - a*b + sqrt(a*b
)*(a - b))*b^7)*abs(-a*b^2 + b^3)*abs(-a + b) - (9*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^5*b^4 - 51*
sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^4*b^5 + 102*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^3*
b^6 - 82*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*a^2*b^7 + 17*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a
*b)*a*b^8 + 5*sqrt(a^2 - a*b + sqrt(a*b)*(a - b))*sqrt(a*b)*b^9)*abs(-a + b))*(pi*floor((d*x + c)/pi + 1/2) +
arctan(tan(d*x + c)/sqrt((a^2*b^2 - a*b^3 - sqrt((a^2*b^2 - a*b^3)^2 - (a^2*b^2 - a*b^3)*(a^2*b^2 - 2*a*b^3 +
b^4)))/(a^2*b^2 - 2*a*b^3 + b^4))))/((3*a^7*b^4 - 21*a^6*b^5 + 59*a^5*b^6 - 85*a^4*b^7 + 65*a^3*b^8 - 23*a^2*b
^9 + a*b^10 + b^11)*abs(-a*b^2 + b^3)) - 2*(2*a*tan(d*x + c)^3 + a*tan(d*x + c))/((a*tan(d*x + c)^4 - b*tan(d*
x + c)^4 + 2*a*tan(d*x + c)^2 + a)*(a*b - b^2)) + 8*(d*x + c)/b^2)/d

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maple [B]  time = 0.31, size = 644, normalized size = 2.01 \[ \frac {\arctan \left (\tan \left (d x +c \right )\right )}{d \,b^{2}}-\frac {a \left (\tan ^{3}\left (d x +c \right )\right )}{2 d b \left (\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right ) \left (a -b \right )}-\frac {a \tan \left (d x +c \right )}{4 d b \left (\left (\tan ^{4}\left (d x +c \right )\right ) a -\left (\tan ^{4}\left (d x +c \right )\right ) b +2 a \left (\tan ^{2}\left (d x +c \right )\right )+a \right ) \left (a -b \right )}-\frac {a^{2} \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{2 d \,b^{2} \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {3 a \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{4 d b \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}+\frac {3 a^{2} \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{8 d b \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {5 a \arctanh \left (\frac {\left (-a +b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}\right )}{8 d \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}-a \right ) \left (a -b \right )}}-\frac {a^{2} \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{2 d \,b^{2} \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {3 a \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{4 d b \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}-\frac {3 a^{2} \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{8 d b \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}+\frac {5 a \arctan \left (\frac {\left (a -b \right ) \tan \left (d x +c \right )}{\sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}}\right )}{8 d \sqrt {a b}\, \left (a -b \right ) \sqrt {\left (\sqrt {a b}+a \right ) \left (a -b \right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^2,x)

[Out]

1/d/b^2*arctan(tan(d*x+c))-1/2/d*a/b/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)/(a-b)*tan(d*x+c)^3-1/4
/d*a/b/(tan(d*x+c)^4*a-tan(d*x+c)^4*b+2*a*tan(d*x+c)^2+a)/(a-b)*tan(d*x+c)-1/2/d*a^2/b^2/(a-b)/(((a*b)^(1/2)-a
)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+3/4/d*a/b/(a-b)/(((a*b)^(1/2)-a)*(a-b)
)^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))+3/8/d*a^2/b/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)-a
)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))-5/8/d*a/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2
)-a)*(a-b))^(1/2)*arctanh((-a+b)*tan(d*x+c)/(((a*b)^(1/2)-a)*(a-b))^(1/2))-1/2/d*a^2/b^2/(a-b)/(((a*b)^(1/2)+a
)*(a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))+3/4/d*a/b/(a-b)/(((a*b)^(1/2)+a)*(a-b))^
(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))-3/8/d*a^2/b/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(a
-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))+5/8/d*a/(a*b)^(1/2)/(a-b)/(((a*b)^(1/2)+a)*(
a-b))^(1/2)*arctan((a-b)*tan(d*x+c)/(((a*b)^(1/2)+a)*(a-b))^(1/2))

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)^8/(a-b*sin(d*x+c)^4)^2,x, algorithm="maxima")

[Out]

Timed out

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mupad [B]  time = 17.60, size = 7640, normalized size = 23.88 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sin(c + d*x)^8/(a - b*sin(c + d*x)^4)^2,x)

[Out]

(atan(((((5120*a^2*b^7 - 17664*a^3*b^6 + 26688*a^4*b^5 - 16320*a^5*b^4 + 3072*a^6*b^3)/(256*(a*b^5 - b^6)) + (
((20480*a^2*b^11 - 110592*a^3*b^10 + 208896*a^4*b^9 - 167936*a^5*b^8 + 49152*a^6*b^7)/(256*(a*b^5 - b^6)) - (t
an(c + d*x)*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)
^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2)*(98304*a^2*b^12 - 196608*a^3*b^11 + 196608*a^5*b^
9 - 98304*a^6*b^8))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5
- 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) - (tan(c + d*x)*(213
76*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 18432*a^6*b^4))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b
^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 -
b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 1
6*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) + (tan(c + d*x)*(768*a^
6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1
/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8))
)^(1/2)*1i - (((5120*a^2*b^7 - 17664*a^3*b^6 + 26688*a^4*b^5 - 16320*a^5*b^4 + 3072*a^6*b^3)/(256*(a*b^5 - b^6
)) + (((20480*a^2*b^11 - 110592*a^3*b^10 + 208896*a^4*b^9 - 167936*a^5*b^8 + 49152*a^6*b^7)/(256*(a*b^5 - b^6)
) + (tan(c + d*x)*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(
a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2)*(98304*a^2*b^12 - 196608*a^3*b^11 + 196608*
a^5*b^9 - 98304*a^6*b^8))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^
2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) + (tan(c + d*x
)*(21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 18432*a^6*b^4))/(128*(a*b^4 - b^5)))*((8*a^
2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*
b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b
^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) - (tan(c + d*x)*(
768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b
^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3
*b^8)))^(1/2)*1i)/((((5120*a^2*b^7 - 17664*a^3*b^6 + 26688*a^4*b^5 - 16320*a^5*b^4 + 3072*a^6*b^3)/(256*(a*b^5
- b^6)) + (((20480*a^2*b^11 - 110592*a^3*b^10 + 208896*a^4*b^9 - 167936*a^5*b^8 + 49152*a^6*b^7)/(256*(a*b^5
- b^6)) - (tan(c + d*x)*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29
*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2)*(98304*a^2*b^12 - 196608*a^3*b^11 + 1
96608*a^5*b^9 - 98304*a^6*b^8))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 +
47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) - (tan(c
+ d*x)*(21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 18432*a^6*b^4))/(128*(a*b^4 - b^5)))*
((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256
*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47
*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) + (tan(c +
d*x)*(768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b^9)^(1/2) + 25*b^
2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9
+ a^3*b^8)))^(1/2) + (((5120*a^2*b^7 - 17664*a^3*b^6 + 26688*a^4*b^5 - 16320*a^5*b^4 + 3072*a^6*b^3)/(256*(a*
b^5 - b^6)) + (((20480*a^2*b^11 - 110592*a^3*b^10 + 208896*a^4*b^9 - 167936*a^5*b^8 + 49152*a^6*b^7)/(256*(a*b
^5 - b^6)) + (tan(c + d*x)*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 -
29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2)*(98304*a^2*b^12 - 196608*a^3*b^11
+ 196608*a^5*b^9 - 98304*a^6*b^8))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^
6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) + (ta
n(c + d*x)*(21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 18432*a^6*b^4))/(128*(a*b^4 - b^5)
))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(
256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2))*((8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) - 35*a*b^6 +
47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) - (tan(c
+ d*x)*(768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2))/(128*(a*b^4 - b^5)))*((8*a^2*(a*b^9)^(1/2) + 25
*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*
b^9 + a^3*b^8)))^(1/2) - (656*a^5 - 2049*a^4*b + 1600*a^3*b^2)/(128*(a*b^5 - b^6))))*((8*a^2*(a*b^9)^(1/2) + 2
5*b^2*(a*b^9)^(1/2) - 35*a*b^6 + 47*a^2*b^5 - 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2
*b^9 + a^3*b^8)))^(1/2)*2i)/d + (atan(((((5120*a^2*b^7 - 17664*a^3*b^6 + 26688*a^4*b^5 - 16320*a^5*b^4 + 3072*
a^6*b^3)/(256*(a*b^5 - b^6)) + (((20480*a^2*b^11 - 110592*a^3*b^10 + 208896*a^4*b^9 - 167936*a^5*b^8 + 49152*a
^6*b^7)/(256*(a*b^5 - b^6)) - (tan(c + d*x)*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*
b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2)*(98304*a^2*b^12
- 196608*a^3*b^11 + 196608*a^5*b^9 - 98304*a^6*b^8))/(128*(a*b^4 - b^5)))*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b
^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3
*b^8)))^(1/2) - (tan(c + d*x)*(21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 18432*a^6*b^4))
/(128*(a*b^4 - b^5)))*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*
a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2))*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^
9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*
b^8)))^(1/2) + (tan(c + d*x)*(768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2))/(128*(a*b^4 - b^5)))*(-(8*
a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*
a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2)*1i - (((5120*a^2*b^7 - 17664*a^3*b^6 + 26688*a^4*b^5 - 16320*a^5*
b^4 + 3072*a^6*b^3)/(256*(a*b^5 - b^6)) + (((20480*a^2*b^11 - 110592*a^3*b^10 + 208896*a^4*b^9 - 167936*a^5*b^
8 + 49152*a^6*b^7)/(256*(a*b^5 - b^6)) + (tan(c + d*x)*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^
6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2)*(9830
4*a^2*b^12 - 196608*a^3*b^11 + 196608*a^5*b^9 - 98304*a^6*b^8))/(128*(a*b^4 - b^5)))*(-(8*a^2*(a*b^9)^(1/2) +
25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^
2*b^9 + a^3*b^8)))^(1/2) + (tan(c + d*x)*(21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 1843
2*a^6*b^4))/(128*(a*b^4 - b^5)))*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^
3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2))*(-(8*a^2*(a*b^9)^(1/2) + 2
5*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2
*b^9 + a^3*b^8)))^(1/2) - (tan(c + d*x)*(768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2))/(128*(a*b^4 - b
^5)))*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2
))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2)*1i)/((((5120*a^2*b^7 - 17664*a^3*b^6 + 26688*a^4*b^5 -
16320*a^5*b^4 + 3072*a^6*b^3)/(256*(a*b^5 - b^6)) + (((20480*a^2*b^11 - 110592*a^3*b^10 + 208896*a^4*b^9 - 16
7936*a^5*b^8 + 49152*a^6*b^7)/(256*(a*b^5 - b^6)) - (tan(c + d*x)*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2
) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^
(1/2)*(98304*a^2*b^12 - 196608*a^3*b^11 + 196608*a^5*b^9 - 98304*a^6*b^8))/(128*(a*b^4 - b^5)))*(-(8*a^2*(a*b^
9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 -
b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) - (tan(c + d*x)*(21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5
*b^5 - 18432*a^6*b^4))/(128*(a*b^4 - b^5)))*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*
b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2))*(-(8*a^2*(a*b^9
)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b
^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) + (tan(c + d*x)*(768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2))/(128
*(a*b^4 - b^5)))*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(
a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) + (((5120*a^2*b^7 - 17664*a^3*b^6 + 26688*a
^4*b^5 - 16320*a^5*b^4 + 3072*a^6*b^3)/(256*(a*b^5 - b^6)) + (((20480*a^2*b^11 - 110592*a^3*b^10 + 208896*a^4*
b^9 - 167936*a^5*b^8 + 49152*a^6*b^7)/(256*(a*b^5 - b^6)) + (tan(c + d*x)*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b
^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3
*b^8)))^(1/2)*(98304*a^2*b^12 - 196608*a^3*b^11 + 196608*a^5*b^9 - 98304*a^6*b^8))/(128*(a*b^4 - b^5)))*(-(8*a
^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a
*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) + (tan(c + d*x)*(21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 2
0864*a^5*b^5 - 18432*a^6*b^4))/(128*(a*b^4 - b^5)))*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 -
47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2))*(-(8*a^
2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 - 29*a*b*(a*b^9)^(1/2))/(256*(3*a*
b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) - (tan(c + d*x)*(768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^
2))/(128*(a*b^4 - b^5)))*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4 -
29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2) - (656*a^5 - 2049*a^4*b + 1600*a^3*
b^2)/(128*(a*b^5 - b^6))))*(-(8*a^2*(a*b^9)^(1/2) + 25*b^2*(a*b^9)^(1/2) + 35*a*b^6 - 47*a^2*b^5 + 16*a^3*b^4
- 29*a*b*(a*b^9)^(1/2))/(256*(3*a*b^10 - b^11 - 3*a^2*b^9 + a^3*b^8)))^(1/2)*2i)/d - atan((((((((((80*a^2*b^11
- 432*a^3*b^10 + 816*a^4*b^9 - 656*a^5*b^8 + 192*a^6*b^7)*1i)/(2*(a*b^5 - b^6)) - (tan(c + d*x)*(98304*a^2*b^
12 - 196608*a^3*b^11 + 196608*a^5*b^9 - 98304*a^6*b^8))/(512*b^2*(a*b^4 - b^5)))*1i)/(2*b^2) + (tan(c + d*x)*(
21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 18432*a^6*b^4)*1i)/(256*(a*b^4 - b^5)))*1i)/(2
*b^2) + ((20*a^2*b^7 - 69*a^3*b^6 + (417*a^4*b^5)/4 - (255*a^5*b^4)/4 + 12*a^6*b^3)*1i)/(2*(a*b^5 - b^6)))/(2*
b^2) - (tan(c + d*x)*(768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2))/(256*(a*b^4 - b^5)))/b^2 - (((((((
(80*a^2*b^11 - 432*a^3*b^10 + 816*a^4*b^9 - 656*a^5*b^8 + 192*a^6*b^7)*1i)/(2*(a*b^5 - b^6)) + (tan(c + d*x)*(
98304*a^2*b^12 - 196608*a^3*b^11 + 196608*a^5*b^9 - 98304*a^6*b^8))/(512*b^2*(a*b^4 - b^5)))*1i)/(2*b^2) - (ta
n(c + d*x)*(21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 18432*a^6*b^4)*1i)/(256*(a*b^4 - b
^5)))*1i)/(2*b^2) + ((20*a^2*b^7 - 69*a^3*b^6 + (417*a^4*b^5)/4 - (255*a^5*b^4)/4 + 12*a^6*b^3)*1i)/(2*(a*b^5
- b^6)))/(2*b^2) + (tan(c + d*x)*(768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2))/(256*(a*b^4 - b^5)))/b
^2)/((((((((((80*a^2*b^11 - 432*a^3*b^10 + 816*a^4*b^9 - 656*a^5*b^8 + 192*a^6*b^7)*1i)/(2*(a*b^5 - b^6)) - (t
an(c + d*x)*(98304*a^2*b^12 - 196608*a^3*b^11 + 196608*a^5*b^9 - 98304*a^6*b^8))/(512*b^2*(a*b^4 - b^5)))*1i)/
(2*b^2) + (tan(c + d*x)*(21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 18432*a^6*b^4)*1i)/(2
56*(a*b^4 - b^5)))*1i)/(2*b^2) + ((20*a^2*b^7 - 69*a^3*b^6 + (417*a^4*b^5)/4 - (255*a^5*b^4)/4 + 12*a^6*b^3)*1
i)/(2*(a*b^5 - b^6)))*1i)/(2*b^2) - (tan(c + d*x)*(768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 - 5295*a^4*b^2)*1i)/(2
56*(a*b^4 - b^5)))/b^2 + (((((((((80*a^2*b^11 - 432*a^3*b^10 + 816*a^4*b^9 - 656*a^5*b^8 + 192*a^6*b^7)*1i)/(2
*(a*b^5 - b^6)) + (tan(c + d*x)*(98304*a^2*b^12 - 196608*a^3*b^11 + 196608*a^5*b^9 - 98304*a^6*b^8))/(512*b^2*
(a*b^4 - b^5)))*1i)/(2*b^2) - (tan(c + d*x)*(21376*a^2*b^8 - 84864*a^3*b^7 + 54912*a^4*b^6 + 20864*a^5*b^5 - 1
8432*a^6*b^4)*1i)/(256*(a*b^4 - b^5)))*1i)/(2*b^2) + ((20*a^2*b^7 - 69*a^3*b^6 + (417*a^4*b^5)/4 - (255*a^5*b^
4)/4 + 12*a^6*b^3)*1i)/(2*(a*b^5 - b^6)))*1i)/(2*b^2) + (tan(c + d*x)*(768*a^6 + 800*a^2*b^4 + 4832*a^3*b^3 -
5295*a^4*b^2)*1i)/(256*(a*b^4 - b^5)))/b^2 - ((41*a^5)/8 - (2049*a^4*b)/128 + (25*a^3*b^2)/2)/(a*b^5 - b^6)))/
(b^2*d) - ((a*tan(c + d*x)^3)/(2*b*(a - b)) + (a*tan(c + d*x))/(4*b*(a - b)))/(d*(a + 2*a*tan(c + d*x)^2 + tan
(c + d*x)^4*(a - b)))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sin(d*x+c)**8/(a-b*sin(d*x+c)**4)**2,x)

[Out]

Timed out

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